Evaluation G - 7 | 1.1 Solutions 1.1.1 What is the place value of the digit 7 in the number 3,705,219? Solution: Number: 3,705,219 Digit 7 is in the hundred-thousands place. Place value = $7 times 10^5$ Answer: $7 times 10^5$ 1.1.2 Which of these numbers, when written in words, contains exactly one 'hundred'? (A) 1,200,345 (B) 105,670 (C) 99,999 (D) 1,001,001 (E) 200,200. Solution: Write each in words: A) 1,200,345 → "one million two hundred thousand three hundred forty-five" → contains "hundred" twice B) 105,670 → "one hundred five thousand six hundred seventy" → contains "hundred" twice C) 99,999 → "ninety-nine thousand nine hundred ninety-nine" → contains "hundred" once ✓ D) 1,001,001 → "one million one thousand one" → contains no "hundred" E) 200,200 → "two hundred thousand two hundred" → contains "hundred" twice Only C (99,999) contains exactly one 'hundred'. Answer: 99,999 1.1.3 If you write "four hundred and eight thousand and seventy" in figures, which digit is in the hundreds place? Solution: "four hundred and eight thousand and seventy" = 408,070 Hundreds place digit = 0 Answer: 0 1.1.4 The distance from Lagos to Abuja is approximately nine hundred and thirty-nine thousand, seven hundred meters. What is this number in figures? Solution: "nine hundred and thirty-nine thousand, seven hundred" = 939,700 Answer: 939,700 1.1.5 What is $300,000 + 5,000 + 40 + 7$ in standard form? Solution: $300,000 + 5,000 + 40 + 7 = 305,047$ Answer: 305,047 1.1.6 Which number has a 9 in the ten-thousands place and a 2 in the thousands place? Solution: Check each option: A) 192,837 → ten-thousands digit = 9 ✓, thousands digit = 2 ✓ B) 902,145 → thousands digit = 2 ✗ C) 591,230 → ten-thousands digit = 9 ✓, thousands digit = 1 ✗ D) 439,210 → ten-thousands digit = 3 ✗ E) 109,235 → ten-thousands digit = 0 ✗ Answer: 192,837 1.1.7 Express $0.45$ as a fraction and find the sum of its numerator and denominator. Solution: $0.45 = frac{45}{100} = frac{9}{20}$ Numerator = 9, Denominator = 20 Sum = $9 + 20 = 29$ Answer: 29 1.1.8 The decimal number 15.083 can be written in expanded form as: Solution: $15.083 = 10 + 5 + frac{8}{100} + frac{3}{1000}$ Answer: $10 + 5 + frac{8}{100} + frac{3}{1000}$ 1.1.9 What is the value of the digit 6 in the number 2.064? Solution: 2.064 = 2 + $frac{0}{10}$ + $frac{6}{100}$ + $frac{4}{1000}$ Digit 6 is in the hundredths place: $6 times frac{1}{100}$ Answer: $6 times frac{1}{100}$ 1.1.10 A company's profit was ₦1,507,892. Which statement about writing this number in words is not correct? Solution: Number: 1,507,892 In words: "One million five hundred seven thousand eight hundred ninety-two" Check each statement: D) Contains 'thousand' twice → ✗ Only appears once ("five hundred seven thousand") The statement that is NOT correct is D (it contains 'thousand' only once, not twice). Answer: D 1.1.11 Round 899,501 to the nearest thousand. Solution: 899,501 → Look at hundreds digit (5), which is 5 or greater, so round up. 899,501 ≈ 900,000 Answer: 900,000 1.1.12 What number is 10,000 less than one million? Solution: One million = 1,000,000 1,000,000 - 10,000 = 990,000 Answer: 990,000 1.1.13 If the digit 5 in a certain number represents 500,000, which of the following could be the number? Solution: Digit 5 represents 500,000 → 5 is in hundred-thousands place. Check options: A) 5,723,419 → 5 is in millions place ✗ B) 2,567,891 → hundred-thousands digit = 5 ✓ C) 7,259,031 → ten-thousands digit = 5 ✗ D) 9,831,450 → units digit = 5 ✗ E) 4,385,210 → thousands digit = 5 ✗ Answer: 2,567,891 1.1.14 The decimal 0.025 is equivalent to: Solution: $0.025 = frac{25}{1000} = frac{1}{40}$ Answer: $frac{25}{1000}$ 1.1.15 Which of these numbers, when read aloud, requires saying "and" for the decimal point? Solution: Numbers with decimal points are read with "and" between whole number and decimal part. Example: 1,000.01 → "one thousand and one hundredth" uses "and". Answer: 1,000.01 1.1.16 What is the difference between the place value of the 1st and last 3s in 303,303? Solution: Number: 303,303 First 3 (from left) = 300,000 Last 3 (from left, ones place) = 3 Difference = 300,000 - 3 = 299,997 Answer: 299,997 1.1.17 If one book costs ₦1,250.75, what is the cost of 100 such books? Solution: Cost of 100 books = $1,250.75 times 100 = 125,075$ Answer: ₦125,075.00 1.1.18 Which of these numbers has the greatest digit in the hundred-thousands place? Solution: All are 6-digit numbers, so first digit is hundred-thousands place. A) 456,789 → 4 B) 654,321 → 6 C) 345,678 → 3 D) 987,654 → 9 ✓ E) 123,456 → 1 Answer: 987,654 1.1.19 Write "seven hundred twenty-five and thirty-four thousandths" in standard decimal form. Solution: "seven hundred twenty-five" = 725 "thirty-four thousandths" = 0.034 Combined: 725.034 Answer: 725.034 1.1.20 The population of a city is 3,363,891. How does the digit in the ten-thousands place compare to the digit in the thousands place? (A) It is 2 times greater. (B) It is 3 more. (C) It is 3 less. (D) They are the same. (E) It is 10 times greater. Solution: Number: 3,363,891 Breakdown: 3,000,000 + 300,000 + 60,000 + 3,000 + 800 + 90 + 1 - Ten-thousands place: 60,000 → digit = 6 - Thousands place: 3,000 → digit = 3 Comparison: 6 vs 3 6 is 2 times 3 (6 ÷ 3 = 2) → "It is 2 times greater" ✓ Answer: It is 2 times greater Evaluation G - 7 | 1.2 Solutions 1.2.1 What is the value of $(-8) times (-4) + 12 div (-3) - (-5)^2$? Solution: $(-8) times (-4) = 32$ $12 div (-3) = -4$ $(-5)^2 = 25$ $32 + (-4) - 25 = 32 - 4 - 25 = 3$ Answer: 3 1.2.2 If $x = -3$ and $y = 2$, evaluate: $2x^2 - 3xy + y^2$ Solution: $x^2 = (-3)^2 = 9$ $2x^2 = 2 times 9 = 18$ $-3xy = -3 times (-3) times 2 = 18$ $y^2 = 2^2 = 4$ $18 + 18 + 4 = 40$ Answer: 40 1.2.3 Which expression has the smallest value? Solution: Calculate each: A) $(-5)^3 = -125$ B) $-125 div 5 = -25$ C) $(-10) times 3 = -30$ D) $(-2)^4 - 50 = 16 - 50 = -34$ E) $(-7) times 4 - (-3) = -28 + 3 = -25$ Smallest: $-125$ (A) Answer: $(-5)^3$ 1.2.4 A submarine at $-350$ m rises $120$ m, then dives $80$ m, then rises $50$ m. What is its final position? Solution: Start: $-350$ Rise 120: $-350 + 120 = -230$ Dive 80: $-230 - 80 = -310$ Rise 50: $-310 + 50 = -260$ Answer: $-260$ m 1.2.5 Simplify: $(-18) div 3 times (-2) + 5 - (-4)^2$ Solution: $(-18) div 3 = -6$ $-6 times (-2) = 12$ $(-4)^2 = 16$ $12 + 5 - 16 = 1$ Answer: 1 1.2.6 Find the product: $(-1)^{10} times (-2)^3 times (-3)^2$ Solution: $(-1)^{10} = 1$ $(-2)^3 = -8$ $(-3)^2 = 9$ $1 times (-8) times 9 = -72$ Answer: $-72$ 1.2.7 If $a = -5$, $b = 3$, and $c = -2$, evaluate: $a^2b - bc^2 + abc$ Solution: $a^2b = (-5)^2 times 3 = 25 times 3 = 75$ $bc^2 = 3 times (-2)^2 = 3 times 4 = 12$ $abc = (-5) times 3 times (-2) = 30$ $75 - 12 + 30 = 93$ Answer: 93 1.2.8 What is the result of $(-12) + (-8) - (-15) + 7 - 9$? Solution: $(-12) + (-8) = -20$ $-20 - (-15) = -20 + 15 = -5$ $-5 + 7 = 2$ $2 - 9 = -7$ Answer: $-7$ 1.2.9 Which is equivalent to $(-24) div (-6) times 2 - (-3)^2$? Solution: $(-24) div (-6) = 4$ $4 times 2 = 8$ $(-3)^2 = 9$ $8 - 9 = -1$ Answer: $-1$ 1.2.10 The temperature was $-5^circ C$. It dropped $8^circ C$, then rose $12^circ C$, then dropped $3^circ C$. What is the final temperature? Solution: Start: $-5$ Drop 8: $-5 - 8 = -13$ Rise 12: $-13 + 12 = -1$ Drop 3: $-1 - 3 = -4$ Answer: $-4^circ C$ 1.2.11 Evaluate: $|-7 + 3| - |5 - 9| + (-2) times (-4)$ Solution: $|-7 + 3| = |-4| = 4$ $|5 - 9| = |-4| = 4$ $(-2) times (-4) = 8$ $4 - 4 + 8 = 8$ Answer: 8 1.2.12 Find the missing number: $(-15) times square = 60$ Solution: $square = 60 div (-15) = -4$ Check: $(-15) times (-4) = 60$ ✓ Answer: $-4$ 1.2.13 Simplify: $(-1)^{99} + (-1)^{100} + (-1)^{101}$ Solution: $(-1)^{99} = -1$ (odd power) $(-1)^{100} = 1$ (even power) $(-1)^{101} = -1$ (odd power) $-1 + 1 + (-1) = -1$ Answer: $-1$ 1.2.14 If $m = -4$ and $n = -1$, find: $(m - n)^2 div (m + n)$ Solution: $m - n = -4 - (-1) = -4 + 1 = -3$ $(m - n)^2 = (-3)^2 = 9$ $m + n = -4 + (-1) = -5$ $9 div (-5) = -frac{9}{5} = -1.8$ Answer: $-1.8$ 1.2.15 Which statement is false? Solution: A) $(-3)^4 = 81$ ✓ B) $(-2)^5 = -32$ ✓ C) $(-7)^2 = -49$ ✗ (should be 49) D) $(-1)^{100} = 1$ ✓ E) $(-4)^3 = -64$ ✓ False: C Answer: $(-7)^2 = -49$ 1.2.16 Calculate: $(-10)^3 div (-20) + (-3) times (-4)^2$ Solution: $(-10)^3 = -1000$ $-1000 div (-20) = 50$ $(-4)^2 = 16$ $(-3) times 16 = -48$ $50 + (-48) = 2$ Answer: 2 1.2.17 Solve: $(-2x + 5) = 17$ when $x$ is an integer. Solution: $-2x + 5 = 17$ $-2x = 17 - 5$ $-2x = 12$ $x = 12 div (-2)$ $x = -6$ Answer: $-6$ 1.2.18 Find the sum: $(-20) + 15 + (-10) + 25 + (-5)$ Solution: Group positives: $15 + 25 = 40$ Group negatives: $-20 + (-10) + (-5) = -35$ $40 + (-35) = 5$ Answer: 5 1.2.19 A debt of $120$ is represented as $-120$. If you pay back $45$, then borrow $60$, what is the new balance? Solution: Start: $-120$ Pay back $45$: $-120 + 45 = -75$ Borrow $60$: $-75 - 60 = -135$ Answer: $-135$ 1.2.20 Evaluate: $(-8) - (-12) + (-5) times 3 div (-1)$ Solution: $(-8) - (-12) = -8 + 12 = 4$ $(-5) times 3 = -15$ $-15 div (-1) = 15$ $4 + 15 = 19$ Answer: 19 Evaluation G - 7 | 1.3 Solutions 1.3.1 If the product of two prime numbers is 91, what is their sum? Solution: Find prime factors of 91: $91 = 7 times 13$ (both prime) Sum: $7 + 13 = 20$ Answer: 20 1.3.2 A factory packs oranges into boxes of 6 and 8. What is the smallest number of oranges it could pack so that both box sizes can be used with no oranges left over? Solution: Need LCM of 6 and 8: $6 = 2 times 3$ $8 = 2^3$ LCM $= 2^3 times 3 = 24$ Answer: 24 1.3.3 The LCM of two numbers is 72 and their HCF is 12. If one number is 24, what is the other? Solution: Use: LCM $times$ HCF = product of the two numbers $72 times 12 = 24 times x$ $864 = 24x$ $x = 864 div 24 = 36$ Answer: 36 1.3.4 How many prime numbers between 30 and 60 end with the digit 3? Solution: Primes between 30 and 60: 31, 37, 41, 43, 47, 53, 59 Those ending in 3: 31, 43, 53 → 3 numbers Answer: 3 1.3.5 The number 360 can be expressed as $2^a times 3^b times 5^c$. What is $a + b + c$? Solution: Prime factorize 360: $360 = 2^3 times 3^2 times 5^1$ $a = 3, b = 2, c = 1$ $a + b + c = 3 + 2 + 1 = 6$ Answer: 6 1.3.6 A perfect cube less than 100 is also a perfect square. What is the sum of all such numbers? Solution: Perfect cubes < 100: 1, 8, 27, 64 Check which are also perfect squares: $1 = 1^2$ ✓ $8$ not square ✗ $27$ not square ✗ $64 = 8^2$ ✓ Numbers: 1 and 64 Sum: $1 + 64 = 65$ Answer: 65 1.3.7 What is the smallest whole number that is divisible by all the numbers from 1 to 6? Solution: Need LCM(1,2,3,4,5,6): Prime factors: $1, 2, 3, 2^2, 5, 2 times 3$ LCM = $2^2 times 3 times 5 = 60$ Answer: 60 1.3.8 If $n$ is a whole number, which of these numbers is always even? Solution: Test each expression for parity: A) $n^2 + 3n = n(n+3)$ → If $n$ is even → even; if $n$ is odd → $n+3$ is even → Always even ✓ B) $n^2 + 2n + 1 = (n+1)^2$ → If $n=0$ → $1$ (odd) ✗ C) $n^3 + 2$ → If $n$ is even → even+2=even; if $n$ is odd → odd+2=odd ✗ D) $n^3 + n^2 + 1$ → If $n=0$ → $1$ (odd) ✗ E) $2n^2 + 1$ → $2n^2$ is always even → even+1=always odd ✗ Only A is always even. Answer: A ($n^2 + 3n$) 1.3.9 What is the HCF of the numbers: $2^3 times 3^2 times 5$, $2^2 times 3^4$, and $2^4 times 3 times 7$? Solution: Take minimum exponents for each prime: For 2: min exponent is $2^2$ For 3: min exponent is $3^1$ For 5,7: not in all numbers → exclude HCF = $2^2 times 3^1 = 4 times 3 = 12$ Answer: 12 1.3.10 A number is divisible by 9 and 15. Which of the following is not true? (A) Divisible by 3, (B) Divisible by 5, (C) Divisible by 45, (D) Divisible by 9, (E) Divisible by 30. Solution: If a number is divisible by 9 and 15, then it must be divisible by their LCM. LCM(9, 15) = 45. Therefore, the number must be a multiple of 45. Check each option for a multiple of 45: A) Divisible by 3 → 45 ÷ 3 = 15 ✓ True B) Divisible by 5 → 45 ÷ 5 = 9 ✓ True C) Divisible by 45 → By definition ✓ True D) Divisible by 9 → Given ✓ True E) Divisible by 30 → 45 ÷ 30 = 1.5 ✗ Not true (45 is not divisible by 30) The statement that is not true is E. Answer: E (Divisible by 30) 1.3.11 A number has exactly 5 factors. What could the number be? Solution: Number with exactly 5 factors = $p^4$ where p is prime $p^4$ factors: 1, p, p², p³, p⁴ Check options: $16 = 2^4$ has factors: 1,2,4,8,16 → 5 factors ✓ Others: 27=3³ (4 factors), 49=7² (3 factors), 64=2⁶ (7 factors), 81=3⁴ (5 factors) So both 16 and 81 work. Answer: 16 1.3.12 The sum of the digits of a two-digit number is 9. The number itself is a multiple of 7. What is the number? Solution: Two-digit multiples of 7 with digit sum 9: Multiples of 7: 14,21,28,35,42,49,56,63,70,77,84,91,98 Digit sum = 9: 36? not multiple of 7; 63 → 6+3=9 ✓ Answer: 63 1.3.13 If the HCF of 56 and another number is 14, and their LCM is 168, what is the other number? Solution: Use: HCF $times$ LCM = product of numbers $14 times 168 = 56 times x$ $2352 = 56x$ $x = 2352 div 56 = 42$ Answer: 42 1.3.14 What is the smallest number by which 1080 must be divided so that the quotient is a perfect cube? Solution: Prime factorize 1080: $1080 = 2^3 times 3^3 times 5^1$ For perfect cube, exponents must be multiples of 3. Currently exponents: 3,3,1 → need to remove 5¹ Divide by 5 → quotient = $2^3 times 3^3 = 216$ (perfect cube) Answer: 5 1.3.15 Find the number of multiples of 7 between 50 and 150. Solution: Smallest multiple ≥ 50: $7 times 8 = 56$ Largest multiple ≤ 150: $7 times 21 = 147$ Number of terms: $frac{147 - 56}{7} + 1 = frac{91}{7} + 1 = 13 + 1 = 14$ Answer: 14 1.3.16 Which number is a perfect square, a perfect cube, and also divisible by 8? Solution: A) 16 = $4^2$ ✓, not cube ✗ B) 64 = $8^2$ ✓, $4^3$ ✓, 64 ÷ 8 = 8 ✓ C) 81 = $9^2$ ✓, not cube ✗ D) 125 = $5^3$ ✓, not square ✗ E) 256 = $16^2$ ✓, not cube ✗ Only 64 satisfies all conditions. Answer: B (64) 1.3.17 The product of two numbers is 294. Their HCF is 7. What is their LCM? Solution: Use: Product = HCF $times$ LCM $294 = 7 times text{LCM}$ $text{LCM} = 294 div 7 = 42$ Answer: 42 1.3.18 What is the remainder when the sum of the first 50 prime numbers is divided by 10? Solution: First 50 primes: 2,3,5,7,11,...,229 Sum last digit: 2 is only even prime, rest odd. Sum of first 50 primes = 5117 (known value) $5117 mod 10 = 7$ Answer: 7 1.3.19 A number leaves a remainder of 4 when divided by 6 and a remainder of 5 when divided by 7. What is the smallest such number? Solution: Let number = $n$ $n = 6a + 4 = 7b + 5$ Test small numbers: 40 works: $40 div 6 = 6$ R4, $40 div 7 = 5$ R5 Check options: 40 not in options (23,31,39,47,53) None work exactly. Smallest is 40. 1.3.20 How many two-digit numbers are either prime or have exactly 3 factors? Solution: Two-digit primes: 11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97 → 21 numbers Numbers with exactly 3 factors = $p^2$ where p is prime Two-digit $p^2$: 25 (5²), 49 (7²) → 2 numbers Total = 21 + 2 = 23 Answer: 23 Evaluation G - 7 | 1.4 Solutions 1.4.1 Simplify: $frac{0.75 + 0.overline{6}}{1.25 - 0.4} times frac{8}{15}$ Solution: $0.75 = frac{3}{4}$, $0.overline{6} = frac{2}{3}$ Sum: $frac{3}{4} + frac{2}{3} = frac{9+8}{12} = frac{17}{12}$ $1.25 - 0.4 = 1.25 - 0.4 = 0.85 = frac{85}{100} = frac{17}{20}$ First fraction: $frac{17/12}{17/20} = frac{17}{12} times frac{20}{17} = frac{20}{12} = frac{5}{3}$ Multiply: $frac{5}{3} times frac{8}{15} = frac{40}{45} = frac{8}{9}$ Answer: $frac{8}{9}$ 1.4.2 If $x = 0.1overline{36}$ and $y = frac{3}{22}$, find $x + y$ expressed as a fraction in its simplest form. Solution: $0.1overline{36} = 0.1363636...$ Let $t = 0.1overline{36}$ → $100t = 13.63636...$ Subtract: $100t - t = 13.63636... - 0.13636... = 13.5$ $99t = 13.5$ → $t = frac{13.5}{99} = frac{135}{990} = frac{3}{22}$ So $x = frac{3}{22}$, $y = frac{3}{22}$ $x + y = frac{3}{22} + frac{3}{22} = frac{6}{22} = frac{3}{11}$ Answer: $frac{3}{11}$ 1.4.3 A trader bought 3 different items. The cost of the first was $frac{1}{3}$ of his money, the second cost $40%$ of the remaining money, and the third cost ₦1200, leaving him with nothing. How much money did he start with? Solution: Let starting money = $M$ First item: $frac{1}{3}M$, remaining = $frac{2}{3}M$ Second item: $40%$ of $frac{2}{3}M = frac{2}{5} times frac{2}{3}M = frac{4}{15}M$ Remaining after second: $frac{2}{3}M - frac{4}{15}M = frac{10}{15}M - frac{4}{15}M = frac{6}{15}M = frac{2}{5}M$ Third item = ₦1200 = $frac{2}{5}M$ $M = 1200 times frac{5}{2} = 3000$ Answer: ₦3000 1.4.4 Evaluate: $frac{2frac{1}{3} times 1frac{1}{2}}{0.2 div 0.5} - 0.overline{3}$ Solution: $2frac{1}{3} = frac{7}{3}$, $1frac{1}{2} = frac{3}{2}$ Product: $frac{7}{3} times frac{3}{2} = frac{7}{2}$ $0.2 div 0.5 = 0.4 = frac{2}{5}$ First part: $frac{7/2}{2/5} = frac{7}{2} times frac{5}{2} = frac{35}{4}$ $0.overline{3} = frac{1}{3}$ $frac{35}{4} - frac{1}{3} = frac{105}{12} - frac{4}{12} = frac{101}{12} = 8frac{5}{12}$ Answer: $8frac{5}{12}$ 1.4.5 Which of the following is NOT equivalent to $0.8%$? Solution: $0.8% = 0.008$ Check each: A) $frac{1}{125} = 0.008$ ✓ B) $0.008$ ✓ C) $frac{4}{500} = 0.008$ ✓ D) $frac{8}{1000} = 0.008$ ✓ E) $frac{80}{100} = 0.8$ ≠ 0.008 ✗ Answer: $frac{80}{100}$ 1.4.6 In an election between two candidates, candidate A received $frac{5}{8}$ of the votes. If candidate B received 15,000 votes, and $2%$ of votes were invalid, how many valid votes were cast? Solution: Candidate A: $frac{5}{8}$ of valid votes Candidate B: $frac{3}{8}$ of valid votes = 15,000 Valid votes = $15000 times frac{8}{3} = 40000$ Answer: 40,000 1.4.7 Simplify: $frac{0.125^{-1} + 0.25^{-1}}{(0.5)^{-2}}$ Solution: $0.125^{-1} = frac{1}{0.125} = 8$ $0.25^{-1} = frac{1}{0.25} = 4$ Sum = $8 + 4 = 12$ $(0.5)^{-2} = frac{1}{0.25} = 4$ Result = $frac{12}{4} = 3$ Answer: 3 1.4.8 If $frac{a}{b} = 0.625$ and $a + b = 104$, what is $b - a$? Solution: $0.625 = frac{5}{8}$ → $a = 5k, b = 8k$ $a + b = 13k = 104$ → $k = 8$ $a = 40, b = 64$ $b - a = 64 - 40 = 24$ Answer: 24 1.4.9 A student incorrectly multiplied a number by $frac{2}{5}$ instead of dividing by $frac{2}{5}$. The result he got was 128 less than the correct answer. What was the original number? Solution: Let original number = $N$ Correct answer = $N div frac{2}{5} = N times frac{5}{2} = frac{5N}{2}$ Incorrect answer = $N times frac{2}{5} = frac{2N}{5}$ Difference = $frac{5N}{2} - frac{2N}{5} = 128$ $frac{25N}{10} - frac{4N}{10} = frac{21N}{10} = 128$ $21N = 1280$ → $N = frac{1280}{21} approx 60.95$ Answer: Not an integer anyway. 1.4.10 Which fraction lies exactly midway between $frac{5}{7}$ and $frac{6}{7}$? Solution: Midpoint = average = $frac{frac{5}{7} + frac{6}{7}}{2} = frac{frac{11}{7}}{2} = frac{11}{14}$ Answer: $frac{11}{14}$ 1.4.11 A tank is $frac{2}{5}$ full. After adding 300 liters, it becomes $0.7$ full. What is the capacity of the tank? Solution: Let capacity = $C$ $frac{2}{5}C + 300 = 0.7C$ $0.7 = frac{7}{10}$ $frac{7}{10}C - frac{2}{5}C = 300$ $frac{7}{10}C - frac{4}{10}C = frac{3}{10}C = 300$ $C = 300 times frac{10}{3} = 1000$ Answer: 1000 L 1.4.12 Find the value of $x$ if $frac{0.4overline{6} + 0.5overline{3}}{0.3overline{8}} = x$. Express as a simplified fraction. Solution: $0.4overline{6} = 0.4666... = frac{7}{15}$ $0.5overline{3} = 0.5333... = frac{8}{15}$ Sum = $frac{7}{15} + frac{8}{15} = frac{15}{15} = 1$ $0.3overline{8} = 0.3888... = frac{7}{18}$ $x = frac{1}{7/18} = frac{18}{7}$ Answer: $frac{18}{7}$ 1.4.13 What percentage of $frac{3}{4}$ is $frac{5}{8}$ of 0.96? Solution: $frac{5}{8}$ of 0.96 = $frac{5}{8} times 0.96 = 0.6$ Percentage = $frac{0.6}{3/4} times 100% = 0.6 times frac{4}{3} times 100% = 0.8 times 100% = 80%$ Answer: 80% 1.4.14 If $2^{n+1} - 2^{n-1} = 12$, find $n$. (A) 2 (B) 3 (C) 4 (D) 5 (E) 6. Solution: $2^{n+1} - 2^{n-1} = 12$ Factor out $2^{n-1}$: $2^{n-1}(2^{2} - 1) = 12$ $2^{n-1}(4 - 1) = 12$ $2^{n-1} times 3 = 12$ $2^{n-1} = 12 div 3$ $2^{n-1} = 4$ $2^{n-1} = 2^2$ $n - 1 = 2$ $n = 3$ Answer: 3 1.4.15 A man shared a sum of money among his three children. The first got $0.375$ of the money, the second got ₦12,000, and the third got $frac{5}{16}$ of the money. How much did the first child get? Solution: $0.375 = frac{3}{8}$ Total = $T$ First + third = $frac{3}{8}T + frac{5}{16}T = frac{6}{16}T + frac{5}{16}T = frac{11}{16}T$ Second = ₦12,000 = $T - frac{11}{16}T = frac{5}{16}T$ $T = 12000 times frac{16}{5} = 38400$ First child = $frac{3}{8} times 38400 = 14400$ Answer: ₦14,400 1.4.16 If $x = 1 + frac{1}{1 + frac{1}{1+0.5}}$, find the value of $x$ as a decimal. Solution: $1 + 0.5 = 1.5$ $1 + frac{1}{1.5} = 1 + frac{2}{3} = frac{5}{3}$ $1 + frac{1}{5/3} = 1 + frac{3}{5} = frac{8}{5} = 1.6$ Answer: 1.6 1.4.17 Find the missing number in the sequence: $frac{3}{4}, 0.8, frac{5}{6}, 0.overline{85}, ?$ Solution: Convert to decimals: $0.75, 0.8, 0.833..., 0.858...$ Pattern as fractions: $frac{3}{4}, frac{4}{5}, frac{5}{6}, frac{6}{7}$ → next would be $frac{7}{8} = 0.875$ Answer: $0.875$ or $frac{7}{8}$ 1.4.18 When a number is increased by $20%$ and then decreased by $25%$, the result is 180. What is the original number? Solution: Let original = $N$ After 20% increase: $1.2N$ After 25% decrease: $0.75 times 1.2N = 0.9N$ $0.9N = 180$ → $N = 200$ Answer: 200 1.4.19 Evaluate: $(0.04)^{-1.5} + (0.125)^{-frac{2}{3}}$ Solution: $(0.04)^{-1.5} = (0.04)^{-frac{3}{2}} = left(frac{1}{25}right)^{-frac{3}{2}} = 25^{frac{3}{2}} = 125$ $(0.125)^{-frac{2}{3}} = left(frac{1}{8}right)^{-frac{2}{3}} = 8^{frac{2}{3}} = 4$ Sum = $125 + 4 = 129$ Answer: 129 1.4.20 A recipe requires $frac{3}{4}$ cup of sugar for every 2 cups of flour. If you want to use 5 cups of flour, and you only have a $frac{1}{3}$ cup measure, how many full measures of sugar are needed? Solution: Sugar per cup of flour = $frac{3/4}{2} = frac{3}{8}$ cup For 5 cups flour: sugar = $5 times frac{3}{8} = frac{15}{8}$ cups Number of $frac{1}{3}$ cup measures = $frac{15/8}{1/3} = frac{15}{8} times 3 = frac{45}{8} = 5.625$ Full measures needed = 5 Answer: 5 Evaluation G - 7 | 1.5 Solutions 1.5.1 Which of the following is arranged in descending order? $frac{5}{8}, 0.64, 63%$ Solution: Convert to decimals: $frac{5}{8} = 0.625$, $0.64 = 0.64$, $63% = 0.63$ Descending order: $0.64 > 0.63 > 0.625$ Order: $0.64, 63%, frac{5}{8}$ Answer: $0.64, 63%, frac{5}{8}$ 1.5.2 A shop advertises a discount of 0.3 off the original price. Another shop advertises a discount of $frac{1}{3}$. Which shop offers a larger discount percentage? Solution: $0.3 = 30%$ $frac{1}{3} approx 33.overline{3}%$ $frac{1}{3} > 0.3$ Second shop offers larger discount. Answer: $frac{1}{3}$ 1.5.3 If $frac{2}{5}$ of a number is 40, what is 0.15 of the same number? Solution: Let number = $N$ $frac{2}{5}N = 40$ → $N = 40 times frac{5}{2} = 100$ $0.15 times 100 = 15$ Answer: 15 1.5.4 Which number is exactly halfway between $-frac{3}{4}$ and $0.5$ on a number line? Solution: Midpoint = $frac{-frac{3}{4} + 0.5}{2}$ $-frac{3}{4} = -0.75$ $frac{-0.75 + 0.5}{2} = frac{-0.25}{2} = -0.125$ Answer: $-0.125$ 1.5.5 In an election, Candidate A got $frac{3}{8}$ of the votes, Candidate B got 0.4, and Candidate C got the rest. Which candidate got the most votes? Solution: $frac{3}{8} = 0.375$, $0.4 = 0.400$ Candidate C = $1 - 0.375 - 0.4 = 0.225$ Largest is Candidate B (0.4) Answer: Candidate B 1.5.6 If $x = 0.overline{6}$, $y = frac{2}{3}$, and $z = 66.7%$, which statement is true? Solution: $x = 0.overline{6} = frac{2}{3}$ $y = frac{2}{3}$ so $x = y$ $z = 66.7% = 0.667$ ≈ $0.overline{6}$ but slightly less So $x = y > z$ Answer: $x = y > z$ 1.5.7 A recipe requires $0.overline{3}$ cups of sugar and $frac{7}{20}$ cups of flour. Which ingredient quantity is greater? Solution: $0.overline{3} = frac{1}{3} approx 0.333$ $frac{7}{20} = 0.35$ $0.35 > 0.333$ Flour is greater. Answer: Flour 1.5.8 What is the result of $left(frac{5}{6} - 0.overline{6}right) times 100%$? Solution: $0.overline{6} = frac{2}{3} = frac{4}{6}$ $frac{5}{6} - frac{4}{6} = frac{1}{6}$ $frac{1}{6} times 100% = 16.overline{6}%$ Answer: $16.overline{6}%$ 1.5.9 If prices increase by 12.5% next month, what fraction of the new price represents the original price? Solution: Increase by 12.5% → multiply by $1.125 = frac{9}{8}$ Original/New = $frac{1}{1.125} = frac{8}{9}$ Answer: $frac{8}{9}$ 1.5.10 Arrange from smallest to largest: $frac{11}{15}, 0.73, 73.overline{3}%$ Solution: $frac{11}{15} = 0.7333...$ $0.73 = 0.7300$ $73.overline{3}% = 0.7333...$ (same as $frac{11}{15}$) Order: $0.73, frac{11}{15}, 73.overline{3}%$ Answer: $0.73, frac{11}{15}, 73.overline{3}%$ 1.5.11 A tank is filled to $frac{7}{12}$ of its capacity. After adding 120 liters, it becomes 0.9 full. What is the full capacity in liters? Solution: Let capacity = $C$ $frac{7}{12}C + 120 = 0.9C$ $0.9 = frac{9}{10}$ $frac{9}{10}C - frac{7}{12}C = 120$ $frac{54}{60}C - frac{35}{60}C = frac{19}{60}C = 120$ $C = 120 times frac{60}{19} approx 378.95$ Answer: Approximately 378.95 liters 1.5.12 Which of these fractions, when converted to a decimal, gives a repeating sequence of 2 digits? Solution: Check each: $frac{1}{7} = 0.overline{142857}$ (6 digits) $frac{1}{8} = 0.125$ (terminating) $frac{1}{9} = 0.overline{1}$ (1 digit) $frac{1}{11} = 0.overline{09}$ (2 digits) ✓ $frac{1}{12} = 0.08overline{3}$ (1 digit) Answer: $frac{1}{11}$ 1.5.13 A student incorrectly calculated $frac{3}{5}$ as 0.35. What is the percentage error? Solution: Correct value = $0.6$ Incorrect value = $0.35$ Absolute error = $0.6 - 0.35 = 0.25$ Percentage error = $frac{0.25}{0.6} times 100% approx 41.67%$ Answer: $41.67%$ 1.5.14 If $frac{a}{b} = 0.overline{142857}$, what is the simplest fraction form of $frac{a}{b}$? Solution: $0.overline{142857}$ is the decimal expansion of $frac{1}{7}$ Answer: $frac{1}{7}$ 1.5.15 Three friends share a pizza. Kemi ate 0.3, Ada ate $frac{5}{16}$, and Bola ate the rest. Who ate the most? Solution: Kemi = $0.3 = 0.300$ Ada = $frac{5}{16} = 0.3125$ Bola = $1 - 0.3 - 0.3125 = 0.3875$ Bola ate the most. Answer: Bola 1.5.16 In a survey, $frac{13}{20}$ of people preferred Option A, 0.62 preferred Option B, and the rest had no preference. Which option was more popular? Solution: Option A = $frac{13}{20} = 0.65$ Option B = $0.62$ $0.65 > 0.62$ Option A is more popular. Answer: Option A 1.5.17 The price of an item is ₦2,400. If a discount of $16frac{2}{3}%$ is applied, what is the new price? Solution: $16frac{2}{3}% = frac{1}{6}$ Discount = $frac{1}{6} times 2400 = 400$ New price = $2400 - 400 = 2000$ Answer: ₦2,000 1.5.18 Which of the following is NOT equivalent to the others? $0.overline{27}, frac{3}{11}, 27.overline{27}%, frac{6}{22}, 0.2727$ Solution: $0.overline{27} = frac{3}{11}$ $frac{3}{11} = 0.overline{27}$ $27.overline{27}% = 0.overline{27}$ $frac{6}{22} = frac{3}{11}$ $0.2727$ (terminating) is not equal to $0.overline{27}$ Answer: $0.2727$ 1.5.19 If $x = frac{2^{-1}}{5^{-2}}$, express $x$ as a percentage. Solution: $x = frac{1/2}{1/25} = frac{1}{2} times 25 = frac{25}{2} = 12.5$ As percentage: $12.5 times 100% = 1250%$ Answer: $1250%$ 1.5.20 On a number line, point P is at $-frac{4}{5}$ and point Q is at $-0.1$. What fraction of the distance from P to Q is the point at $-0.5$? Solution: P = $-0.8$, Q = $-0.1$, distance = $0.7$ Point R = $-0.5$, distance from P to R = $0.3$ Fraction = $frac{0.3}{0.7} = frac{3}{7}$ Answer: $frac{3}{7}$ Evaluation G - 7 | 1.6 Solutions 1.6.1 Which of the following is equal to $frac{7}{8}$? $frac{7}{8} = 7 div 8 = 0.875 = 87.5%$ → Answer: $87.5%$ 1.6.2 A recipe requires $frac{5}{6}$ cup of sugar. What is this as a decimal, rounded to three decimal places? $frac{5}{6} = 0.83333... approx 0.833$ → Answer: $0.833$ 1.6.3 If a test score of $frac{17}{20}$ is converted to a percentage, what is the result? $frac{17}{20} = 0.85 = 85%$ → Answer: $85%$ 1.6.4 Which list is in ascending order? Convert to decimals: $frac{3}{5} = 0.6$, $65% = 0.65$, $0.7$, $frac{4}{5} = 0.8$ → Order: $0.6, 0.65, 0.7, 0.8$ → Answer: $frac{3}{5}, 65%, 0.7, frac{4}{5}$ 1.6.5 A store offers a discount of $12.5%$. What fraction of the original price will you pay? $12.5% = frac{1}{8}$ discount → Pay $1 - frac{1}{8} = frac{7}{8}$ → Answer: $frac{7}{8}$ 1.6.6 The repeating decimal $0.overline{6}$ is equivalent to: $0.overline{6} = frac{2}{3}$ → Answer: $frac{2}{3}$ 1.6.7 What is $frac{9}{11}$ expressed as a percentage, rounded to the nearest tenth? $frac{9}{11} approx 0.8181... = 81.8%$ → Answer: $81.8%$ 1.6.8 A population increased by $frac{1}{4}$. What is the percentage increase? $frac{1}{4} = 0.25 = 25%$ → Answer: $25%$ 1.6.9 Which of these is NOT equivalent to the others? $0.125 = 12.5% = frac{1}{8} = frac{5}{40}$ → $0.overline{125}$ is repeating, not equal → Answer: $0.overline{125}$ 1.6.10 A phone's battery is at $frac{3}{10}$. What is this as a decimal and a percentage? $frac{3}{10} = 0.3 = 30%$ → Answer: $0.3, 30%$ 1.6.11 A shirt costs ₦2,400. A sales tax of $8frac{1}{3}%$ is added. What is the tax amount in Naira? $8frac{1}{3}% = frac{25}{3}% = frac{25}{300} = frac{1}{12}$ → Tax = $2400 times frac{1}{12} = 200$ → Answer: ₦$200$ 1.6.12 What is $0.00overline{36}$ as a fraction in simplest form? Let $x = 0.00overline{36}$ → $100x = 0.overline{36}$ → $100x = frac{36}{99}$ → $x = frac{36}{9900} = frac{1}{275}$ → Answer: $frac{1}{275}$ 1.6.13 If $x = frac{5}{12}$, which expression represents $x$ as a percentage? To convert fraction to %: $(numerator div denominator) times 100$ → $(5 div 12) times 100$ → Answer: $(5 div 12) times 100$ 1.6.14 After a $15%$ pay cut, a worker's salary is ₦42,500. What was the original salary? Let original salary = $S$ → $S times (1 - 0.15) = 42500$ → $S = frac{42500}{0.85} = 50000$ → Answer: ₦$50,000$ 1.6.15 Which fraction is equal to $0.1overline{6}$? $0.1overline{6} = 0.1666... = frac{1}{6}$ → Answer: $frac{1}{6}$ 1.6.16 In a survey, $frac{7}{8}$ of people preferred Option A. What percentage did NOT prefer Option A? Not prefer = $1 - frac{7}{8} = frac{1}{8} = 0.125 = 12.5%$ → Answer: $12.5%$ 1.6.17 The decimal number $0.375$ is equivalent to which of these combinations? $0.375 = frac{3}{8} = 37.5%$ → Answer: $frac{3}{8}, 37.5%$ 1.6.18 A car's fuel tank is $0.overline{3}$ full. What fraction of the tank is empty? Full = $frac{1}{3}$ → Empty = $1 - frac{1}{3} = frac{2}{3}$ → Answer: $frac{2}{3}$ 1.6.19 Which calculation correctly converts $frac{2}{9}$ to a decimal? Fraction to decimal = numerator ÷ denominator = $2 div 9$ → Answer: $2 div 9$ 1.6.20 In a class, $60%$ are girls. What fraction are boys, in simplest form? Boys = $100% - 60% = 40% = frac{40}{100} = frac{2}{5}$ → Answer: $frac{2}{5}$ Evaluation G - 7 | 1.7 Solutions 1.7.1 Calculate: $2frac{3}{4} div 1frac{1}{2} times frac{2}{3}$ Solution: $2frac{3}{4} = frac{11}{4}$, $1frac{1}{2} = frac{3}{2}$ $frac{11}{4} div frac{3}{2} = frac{11}{4} times frac{2}{3} = frac{22}{12} = frac{11}{6}$ $frac{11}{6} times frac{2}{3} = frac{22}{18} = frac{11}{9}$ Answer: $frac{11}{9}$ 1.7.2 Simplify: $(frac{3}{5} + frac{2}{3}) div (frac{4}{7} - frac{1}{3})$ Solution: $frac{3}{5} + frac{2}{3} = frac{9+10}{15} = frac{19}{15}$ $frac{4}{7} - frac{1}{3} = frac{12-7}{21} = frac{5}{21}$ $frac{19}{15} div frac{5}{21} = frac{19}{15} times frac{21}{5} = frac{399}{75} = frac{133}{25}$ Answer: $frac{133}{25}$ 1.7.3 What is $frac{5}{6}$ of $frac{7}{10}$ of $frac{9}{14}$? Solution: $frac{5}{6} times frac{7}{10} times frac{9}{14} = frac{5 times 7 times 9}{6 times 10 times 14} = frac{315}{840} = frac{3}{8}$ Answer: $frac{3}{8}$ 1.7.4 A recipe requires $3frac{1}{3}$ cups of flour. If you want to make $2frac{1}{2}$ batches, how many cups of flour are needed? Solution: $3frac{1}{3} = frac{10}{3}$, $2frac{1}{2} = frac{5}{2}$ $frac{10}{3} times frac{5}{2} = frac{50}{6} = frac{25}{3} = 8frac{1}{3}$ Answer: $8frac{1}{3}$ 1.7.5 Calculate: $(1frac{1}{2})^2 - (frac{2}{3})^2 times 2frac{1}{4}$ Solution: $1frac{1}{2} = frac{3}{2}$, $2frac{1}{4} = frac{9}{4}$ $(frac{3}{2})^2 = frac{9}{4}$, $(frac{2}{3})^2 = frac{4}{9}$ $frac{4}{9} times frac{9}{4} = 1$ $frac{9}{4} - 1 = frac{9}{4} - frac{4}{4} = frac{5}{4}$ Answer: $frac{5}{4}$ 1.7.6 If $x = 1frac{1}{3}$ and $y = 2frac{1}{2}$, find $frac{x + y}{x - y}$ in simplest form. Solution: $x = frac{4}{3}$, $y = frac{5}{2}$ $x+y = frac{4}{3} + frac{5}{2} = frac{8+15}{6} = frac{23}{6}$ $x-y = frac{4}{3} - frac{5}{2} = frac{8-15}{6} = -frac{7}{6}$ $frac{23/6}{-7/6} = frac{23}{-7} = -frac{23}{7}$ Answer: $-frac{23}{7}$ 1.7.7 A water tank is $frac{3}{5}$ full. After adding $12frac{1}{2}$ liters, it becomes $frac{7}{8}$ full. What is the capacity of the tank? Solution: Let capacity = $C$ $frac{3}{5}C + 12.5 = frac{7}{8}C$ $frac{7}{8}C - frac{3}{5}C = 12.5$ $frac{35-24}{40}C = 12.5$ $frac{11}{40}C = 12.5$ $C = 12.5 times frac{40}{11} = frac{500}{11} approx 45.45$ Answer: $45.45$ 1.7.8 Simplify: $frac{2frac{1}{4} times 1frac{1}{3}}{frac{3}{5} div frac{1}{2}}$ Solution: $2frac{1}{4} = frac{9}{4}$, $1frac{1}{3} = frac{4}{3}$ $frac{9}{4} times frac{4}{3} = frac{36}{12} = 3$ $frac{3}{5} div frac{1}{2} = frac{3}{5} times 2 = frac{6}{5}$ $frac{3}{6/5} = 3 times frac{5}{6} = frac{15}{6} = frac{5}{2} = 2frac{1}{2}$ Answer: $2frac{1}{2}$ 1.7.9 Calculate: $frac{1}{1 + frac{1}{1 + frac{1}{2}}}$ Solution: $1 + frac{1}{2} = frac{3}{2}$ $1 + frac{1}{3/2} = 1 + frac{2}{3} = frac{5}{3}$ $frac{1}{5/3} = frac{3}{5}$ Answer: $frac{3}{5}$ 1.7.10 A shop has $12frac{3}{4}$ kg of rice. If each customer buys $frac{3}{8}$ kg, how many complete customers can be served? Solution: $12frac{3}{4} = frac{51}{4}$ $frac{51}{4} div frac{3}{8} = frac{51}{4} times frac{8}{3} = frac{408}{12} = 34$ Answer: $34$ 1.7.11 Find the value of $frac{2}{5} + frac{3}{4} - frac{5}{6} times frac{9}{10}$ Solution: $frac{5}{6} times frac{9}{10} = frac{45}{60} = frac{3}{4}$ $frac{2}{5} + frac{3}{4} - frac{3}{4} = frac{2}{5}$ Answer: $frac{2}{5}$ 1.7.12 If $a = 5frac{1}{4}$ and $b = 3frac{1}{2}$, what is $a^2 - b^2$? Solution: $a = frac{21}{4}$, $b = frac{7}{2} = frac{14}{4}$ $a^2 = frac{441}{16}$, $b^2 = frac{196}{16}$ $a^2 - b^2 = frac{245}{16} = 15frac{5}{16}$ Answer: $15frac{5}{16}$ 1.7.13 A car travels $15frac{3}{4}$ km using $1frac{1}{5}$ liters of petrol. How far can it travel on $4$ liters? Solution: $15frac{3}{4} = frac{63}{4}$, $1frac{1}{5} = frac{6}{5}$ Km per liter = $frac{63/4}{6/5} = frac{63}{4} times frac{5}{6} = frac{315}{24} = frac{105}{8}$ Distance for 4 liters = $4 times frac{105}{8} = frac{420}{8} = 52.5 = 52frac{1}{2}$ Answer: $52frac{1}{2}$ 1.7.14 Simplify: $frac{frac{2}{3} - frac{1}{4}}{frac{3}{5} + frac{1}{2}} div frac{2}{7}$ Solution: $frac{2}{3} - frac{1}{4} = frac{8-3}{12} = frac{5}{12}$ $frac{3}{5} + frac{1}{2} = frac{6+5}{10} = frac{11}{10}$ $frac{5/12}{11/10} = frac{5}{12} times frac{10}{11} = frac{50}{132} = frac{25}{66}$ $frac{25}{66} div frac{2}{7} = frac{25}{66} times frac{7}{2} = frac{175}{132}$ Answer: $frac{175}{132}$ 1.7.15 A student spends $frac{1}{4}$ of her money on books, $frac{1}{3}$ of the remainder on transport, and has ₦2,500 left. How much did she have initially? Solution: Let initial = $x$ After books: $x - frac{x}{4} = frac{3x}{4}$ After transport: $frac{3x}{4} - frac{1}{3} times frac{3x}{4} = frac{3x}{4} - frac{x}{4} = frac{x}{2}$ $frac{x}{2} = 2500$ $x = 5000$ Answer: ₦5000 1.7.16 Evaluate: $(3frac{1}{3} - 1frac{3}{4}) times (2frac{1}{2} + 1frac{1}{6})$ Solution: $3frac{1}{3} - 1frac{3}{4} = frac{10}{3} - frac{7}{4} = frac{40-21}{12} = frac{19}{12}$ $2frac{1}{2} + 1frac{1}{6} = frac{5}{2} + frac{7}{6} = frac{15+7}{6} = frac{22}{6} = frac{11}{3}$ $frac{19}{12} times frac{11}{3} = frac{209}{36} = 5frac{29}{36}$ Answer: $5frac{29}{36}$ 1.7.17 Which is largest? Solution: (A) $frac{5}{8} div frac{2}{3} = frac{15}{16} = 0.9375$ (B) $frac{3}{4} times 1frac{1}{2} = frac{3}{4} times frac{3}{2} = frac{9}{8} = 1.125$ (C) $1frac{1}{3} - frac{5}{12} = frac{4}{3} - frac{5}{12} = frac{16-5}{12} = frac{11}{12} approx 0.9167$ (D) $frac{7}{9} + frac{1}{6} = frac{14+3}{18} = frac{17}{18} approx 0.9444$ (E) $frac{4}{5} times frac{10}{9} = frac{40}{45} = frac{8}{9} approx 0.8889$ Largest is (B) $frac{9}{8}$ Answer: $frac{9}{8}$ 1.7.18 A rope $18frac{2}{3}$ meters long is cut into pieces each $1frac{1}{6}$ meters long. How many complete pieces are obtained? Solution: $18frac{2}{3} = frac{56}{3}$, $1frac{1}{6} = frac{7}{6}$ $frac{56}{3} div frac{7}{6} = frac{56}{3} times frac{6}{7} = frac{336}{21} = 16$ Answer: $16$ 1.7.19 Calculate: $frac{frac{1}{2} + frac{2}{3}}{frac{3}{4} - frac{1}{3}} times frac{5}{6}$ Solution: $frac{1}{2} + frac{2}{3} = frac{3+4}{6} = frac{7}{6}$ $frac{3}{4} - frac{1}{3} = frac{9-4}{12} = frac{5}{12}$ $frac{7/6}{5/12} = frac{7}{6} times frac{12}{5} = frac{84}{30} = frac{14}{5}$ $frac{14}{5} times frac{5}{6} = frac{14}{6} = frac{7}{3} = 2frac{1}{3}$ Answer: $2frac{1}{3}$ 1.7.20 If $frac{2}{5}$ of a number is $3frac{1}{3}$, what is $frac{3}{4}$ of the number? Solution: Let number = $n$ $frac{2}{5}n = frac{10}{3}$ $n = frac{10}{3} times frac{5}{2} = frac{50}{6} = frac{25}{3}$ $frac{3}{4} times frac{25}{3} = frac{25}{4} = 6frac{1}{4}$ Answer: $6frac{1}{4}$ Evaluation G - 7 | 1.8 Solutions 1.8.1 Increase from ₦12,000 to ₦15,000 Solution: Increase $= 15{,}000 - 12{,}000 = 3{,}000$ Percentage increase $= frac{3{,}000}{12{,}000} times 100% = 25%$ Answer: $25%$ 1.8.2 Population decreases from 48,000 to 42,000 Solution: Decrease $= 48{,}000 - 42{,}000 = 6{,}000$ Percentage decrease $= frac{6{,}000}{48{,}000} times 100% = 12.5%$ Answer: $12.5%$ 1.8.3 $15%$ of a number is $360$ Solution: Let the number be $x$ $0.15x = 360$ $x = frac{360}{0.15} = 2{,}400$ Answer: $2{,}400$ 1.8.4 Cost price ₦8,000, loss $12.5%$ Solution: Loss $= 12.5%$ of $8{,}000 = 0.125 times 8{,}000 = 1{,}000$ Selling price $= 8{,}000 - 1{,}000 = 7{,}000$ Answer: ₦7,000 1.8.5 Increase by $10%$, then decrease by $10%$ Solution: Let original price be $100$ After increase: $100 + 10% = 110$ After decrease: $110 - 10%$ of $110 = 110 - 11 = 99$ This is $1%$ less than original Answer: $1%$ less than original 1.8.6 $72%$ of $250$ marks Solution: $0.72 times 250 = 180$ Answer: $180$ 1.8.7 $35%$ on food, $25%$ on rent Solution: Total spent $= 35% + 25% = 60%$ Remaining $= 100% - 60% = 40%$ Answer: $40%$ 1.8.8 Marked $20%$ above cost, discount $10%$ Solution: Let cost price $= 100$ Marked price $= 120$ Selling price $= 120 - 10%$ of $120 = 108$ Gain $= 108 - 100 = 8$ Gain percentage $= 8%$ Answer: $8%$ 1.8.9 $40%$ of class are boys, boys = 18 Solution: Total students $= frac{18}{0.4} = 45$ Answer: $45$ 1.8.10 Increase from $80$ to $100$ Solution: Increase $= 20$ Percentage increase $= frac{20}{80} times 100% = 25%$ Answer: $25%$ 1.8.11 Depreciation $15%$ on ₦2,000,000 Solution: Depreciation $= 0.15 times 2{,}000{,}000 = 300{,}000$ Value after one year $= 2{,}000{,}000 - 300{,}000 = 1{,}700{,}000$ Answer: ₦1,700,000 1.8.12 ₦4,500 after $25%$ discount Solution: Selling price $= 75%$ of marked price Marked price $= frac{4{,}500}{0.75} = 6{,}000$ Answer: ₦6,000 1.8.13 Price increases by $20%$ to ₦3,600 Solution: Original price $= frac{3{,}600}{1.2} = 3{,}000$ Answer: ₦3,000 1.8.14 $60%$ of a number is $420$ Solution: Number $= frac{420}{0.6} = 700$ Answer: $700$ 1.8.15 Gain $15%$, selling price ₦9,200 Solution: Cost price $= frac{9{,}200}{1.15} = 8{,}000$ Answer: ₦8,000 1.8.16 Population from 5,000 to 5,750 Solution: Increase $= 750$ Percentage increase $= frac{750}{5{,}000} times 100% = 15%$ Answer: $15%$ 1.8.17 ₦50,000 increases by $8%$ Solution: Increase $= 0.08 times 50{,}000 = 4{,}000$ New value $= 54{,}000$ Answer: ₦54,000 1.8.18 ₦68,000 after $15%$ discount Solution: Marked price $= frac{68{,}000}{0.85} = 80{,}000$ Answer: ₦80,000 1.8.19 Rice price drops from ₦40,000 to ₦34,000 Solution: Decrease $= 6{,}000$ Percentage decrease $= frac{6{,}000}{40{,}000} times 100% = 15%$ Answer: $15%$ 1.8.20 If the length of a rod is increased by $25%$, and the new length is $75text{ cm}$, what was the original length of the rod? Solution: An increase of $25%$ means the new length is $125%$ of the original length. That is, $125% = frac{125}{100} = 1.25$ Let the original length be $xtext{ cm}$. Then $1.25x = 75$ Divide both sides by $1.25$: $x = frac{75}{1.25}$ $x = 60$ Answer: $60text{ cm}$ Evaluation G - 7 | 1.9 Solutions 1.9.1 Round $4,786$ to the nearest hundred. Solution: Look at the tens digit: 8, which is 5 or greater, so round up. $4,786 → 4,800$ Answer: $4,800$ 1.9.2 What is $6,249$ rounded to the nearest ten? Solution: Look at the units digit: 9, which is 5 or greater, so round up. $6,249 → 6,250$ Answer: $6,250$ 1.9.3 Round $89.6$ to the nearest whole number. Solution: The tenths digit is 6, which is 5 or greater, so round up. $89.6 → 90$ Answer: $90$ 1.9.4 A bag of rice costs ₦$19,760$. Estimate the cost to the nearest thousand. Solution: Look at the hundreds digit: 7, which is 5 or greater, so round up. $19,760 → 20,000$ Answer: ₦$20,000$ 1.9.5 Which is the best estimate of $498 times 62$? Solution: Round $498 → 500$, $62 → 60$ $500 times 60 = 30,000$ Answer: $30,000$ 1.9.6 Round $3.742$ to the nearest whole number. Solution: Look at the tenths digit: 7, which is 5 or greater, so round up. $3.742 → 4$ Answer: $4$ 1.9.7 Which number rounds to $4,000$ when rounded to the nearest thousand? Solution: Numbers from $3,500$ up to but not including $4,500$ round to $4,000$ to the nearest thousand. $3,500$ is the smallest integer that rounds to $4,000$. Answer: $3,500$ 1.9.8 Estimate ₦$2,975 + 4,032$ to the nearest thousand. Solution: Round each: $2,975 → 3,000$, $4,032 → 4,000$ Sum: $3,000 + 4,000 = 7,000$ Answer: ₦$7,000$ 1.9.9 Round $15,449$ to the nearest hundred. Solution: Look at the tens digit: 4, which is less than 5, so round down. $15,449 → 15,400$ Answer: $15,400$ 1.9.10 The exact value of ₦$9,842$ is rounded to the nearest ten. What is the result? Solution: Look at the units digit: 2, which is less than 5, so round down. $9,842 → 9,840$ Answer: ₦$9,840$ 1.9.11 Estimate $802 times 49$ by rounding both numbers. Solution: Round $802 → 800$, $49 → 50$ $800 times 50 = 40,000$ Answer: $40,000$ 1.9.12 Which is closest to the value of $3,498 + 5,503$? Solution: Round each: $3,498 → 3,500$, $5,503 → 5,500$ Sum: $3,500 + 5,500 = 9,000$ Answer: $9,000$ 1.9.13 Round $0.496$ to the nearest whole number. Solution: The tenths digit is 4, which is less than 5, so round down. $0.496 → 0$ Answer: $0$ 1.9.14 Estimate ₦$12,480 - 6,513$ to the nearest hundred. Solution: Round each: $12,480 → 12,500$, $6,513 → 6,500$ Difference: $12,500 - 6,500 = 6,000$ Answer: ₦$6,000$ 1.9.15 Round $74,651$ to the nearest thousand. Solution: Look at the hundreds digit: 6, which is 5 or greater, so round up. $74,651 → 75,000$ Answer: $75,000$ 1.9.16 Which is the best estimate of $1,249 div 4$? Solution: Round $1,249 → 1,200$ $1,200 div 4 = 300$ Answer: $300$ 1.9.17 Round ₦$5,995$ to the nearest hundred. Solution: Look at the tens digit: 9, which is 5 or greater, so round up. $5,995 → 6,000$ Answer: ₦$6,000$ 1.9.18 Estimate $19.8 + 40.3$ to the nearest whole number. Solution: Round each: $19.8 → 20$, $40.3 → 40$ Sum: $20 + 40 = 60$ Answer: $60$ 1.9.19 Which number rounds to $700$ when rounded to the nearest hundred and is not $700$ itself? Solution: Numbers from $650$ to $749$ round to $700$. $749$ is one such number that is not exactly $700$. Answer: $749$ 1.9.20 Estimate ₦$24,980 times 3$ by rounding. Solution: Round $24,980 → 25,000$ $25,000 times 3 = 75,000$ Answer: ₦$75,000$
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